Mathematics & Physics 2016, 9(2), 166–172 УДК 519.682+517.55 On Solvability of Systems of Symbolic Polynomial Equations Oleg I. Egorushkin∗ Irina V.Kolbasina† Konstantin V. Safonov‡ Institute of Computer Science and Telecommunications Reshetnev Siberian State Aerospace University Krasnoyarsky Rabochy, 31, Krasnoyarsk, 660014 Russia Received 20.12.2015, received in revised form 24.01.2016, accepted 02.03.2016 Approaches to solving the systems of non-commutative polynomial equations in the form of formal power series (FPS) based on the relation with the corresponding commutative equations are developed. <...> Every FPS is mapped to its commutative image – power series, which is obtained under the assumption that all symbols of the alphabet denote commutative variables assigned as values in the field of complex numbers. <...> It is proved that if the initial non-commutative system of polynomial equations is consistent, then the system of equations being its commutative image is consistent. <...> It is shown that in the case of a non-commutative ring the system of equations can have no solution, have a finite number of solutions, as well as having an infinite number of solutions, which is fundamentally different from the case of complex variables. <...> Firstly, we can consider the symbols z1, . . . , zn,x1, . . . ,xm as an alphabet, over which noncommutative multiplication (concatenation) and commutative operation of the formal sum are determined; besides, the commutative multiplication by complex numbers is defined. <...> The multiplication of any element by (−1) gives the inverse element with respect to addition, so the alphabet generates the ring of symbolic polynomials and FPS with numeric (complex) coefficients. <...> According to the second interpretation, the symbols z1, . . . , zn,x1, . . . ,xm are treated as variables with values from a ring, in which as usual the operation of addition is commutative, while for the operation of multiplication commutative is not required; it is also assumed that elements ∗olegegoruschkin@yandex.ru †kabaskina@yandex.ru ‡safonovkv@rambler.ru ⃝ Siberian Federal University. <...> All rights reserved c – 166 – Oleg I. Egorushkin, Irina V.Kolbasina, Konstantin V. Safonov On Solvability of Systems of Symbolic . . . of the ring may be multiplied by complex numbers. <...> In this case, the problem of solving the system (1) is to express the matrix variables z1, . . . , zn in the form of FPS depending on the non-commutative matrix variables x1, . . . ,xm. <...> In this interpretation the monomials <...>